bee lecture1

7/29/2019 BEE Lecture1

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16/08/2013 Bahman R. Alyaei 1

Chapter 2

Fundamentals of Electric Circuit

Part 1

Charles Coulomb

(17361806)Gustav Robert Kirchhoff

(18241887)


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Introduction

This chapter presents the fundamentallaws ofCircuit Analysis.

We will define the charge, current,

voltage, and power. We will study the basic laws of electrical

circuit analysis called Kirchhoffs Laws.

The basic circuit elements are introduced(current and voltage source and resistor).

Ohoms Laws.


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2.1 Charge, Current, And

Kirchhoffs Current Law

The building block of a matter is the Atom.

The Atom consists of a Nucleus (Neutrons and

Protons) surrounded by Electrons.

The fundamental electric quantity is Charge. Electron and Proton are called charge-carrying

particle in an atom.

The smallest amount of charge that exists is the

charge carried by an electron or proton, equal to (in

Coulomb)

qe= -1.602 x 10-19 C, qp= - qe


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Electrons and Protons are often referredto as Elementary Charges.

Electric Current: is the time rate of

change passing through a predeterminedarea.

This area is the cross-sectional area of

a metal wire. The unit of current is C/s, orAmperes.

where 1 Ampere = 1 Coulomb/second.


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dt

dqi

t

qi

where we imagine qunits ofcharge flowing through the cross-

sectional area A in tunits of time,

and in differential form is shown

below.


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The EE Convention states that: the positivedirection of current flow is that ofPositiveCharges.

In metallic conductors, however, currentis carried by Negative Charges.

These charges are the free electrons in theconduction band, which are only weakly

attracted to the atomic structure in metallicelements and are therefore easily displacedin the presence of electric fields.


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Ex 2.1: Charge and Current In A

Conductor

Find the total charge carrierQin a

cylindrical conductor (solid wire) and

compute the current Iflowing in the wire if:

1. Conductor length: L = 1 m.

2. Conductor diameter: 2r= 2103 m.

3. Charge density: n = 1029 carriers/m3.

4. Charge of one electron: qe=1.602 10-19.

5. Charge carrier velocity: u = 19.910-6 m/s.


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Now, we move one step forward,

We need a flowing current in a conductor.

In order for current to flow there must exista Closed Circuit.


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A simple electrical circuit,composed of a battery(e.g., a dry-cell or alkaline1.5 V battery) and a lightbulb.

Note that the circuitcurrent, i, flowing from thebattery to the light bulb isequal to the currentflowing from the light bulbto the battery.


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Hence, no current (and therefore no

charge) is lost around the closed circuit.

This principle is known as Kirchhoffs

Current Law (KCL).

KCL states that, because charge cannot

be created but must be conserved, the sum

of the currents at a Node must equal zero.

Node: is the junction of two or more

conductors.


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We usually defines

currents entering a nodeas being Negativeand

currents exiting the node

as being Posi t ive.

Thus, the resulting

expression fornode 1 of

the circuit shown is


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Ex 2.2 KCL Applied To An

Automotive Electrical Harness

Figure 2.4 shows an automotive battery

connected to a variety of circuits in an

automobile.

The circuits include headlights, taillights,starter motor, fan, power locks, and

dashboard panel.

The battery must supply enough current toindependently satisfy the requirements of each

of the Load circuits.

Apply KCL to the automotive circuits.


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(a)


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(c)


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2.2 Voltage And Kirchhoffs

Voltage Law

It must take some Work, orEnergy, forthe charge to move between two points ina circuit, say, from point ato point b.

Voltage: is the total work per unit chargeassociated with the motion of chargebetween two points.

The units of voltage are those ofenergyper unit charge, or simply Volt.

1 Volt = 1 Joule/Coulomb.


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The Voltage, orPotential Difference,between two points in a circuit indicates theenergy required to move charge from onepoint to the other.

The polarity, of the voltage is closely tied towhether energy is being dissipated orgenerated in the process.

No energy is lost or created in an electriccircuit.

This principle is known as KirchhoffsVoltage Law (KVL).


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KVL states that, the sum of all voltages

associated with sources must equal the

sum of the load voltages.

So that the net voltage around a closed

circuit is zero.


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Where the vnare the individual

voltages around the closed

circuit.

For the circuit shown, it mustfollow from KVL that the work

generated by the battery is equal

to the energy dissipated in the

bulb in order to sustain thecurrent flow and to convert the

electric energy to heat and light,

hence or


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One may think of the work done in moving a

charge from point ato point band the work

done moving it back from bto aas

corresponding directly to the voltages acrossindividual circuit elements.

Let Qbe the total charge that moves around the

circuit per unit time, giving rise to the current i.

Then the work done in moving Qfrom bto a

(i.e., across the battery) is


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Similarly, work is done in moving Qfrom a

to b, that is, across the light bulb.

Voltage represents the potential energy

between two points in a circuit.

If we remove the light bulb from its

connections to the battery, there still exists

a voltage across the (now disconnected)

terminals band a.


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The presence of a

voltage, v2, across

the open terminals a

and bindicates thepotential energy that

can enable the motion

of charge.

Once a closed circuitis established to allow

current to flow.


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The battery by virtue of an electrochemicallyinduced separation of charge, a 1.5-Vpotential difference is generated.

The potential generated by the battery maybe used to move charge in a circuit.

The rate at which charge is moved once aclosed circuit is established (i.e., the current

drawn by the circuit connected to the battery)depends now on the circuit element wechoose to connect to the battery.


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Hence, the voltage across the battery

represents the potential for providing energy

to a circuit.

Then, battery is a source of energy.

Also, the voltage across the light bulb

indicates the amount ofwork done in

dissipating energy. In the first case, energy is generated; in

the second, it is consumed.


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This fundamental distinction requires

attention in defining the sign (polarity) of

voltages.

We will refer to elements that provideenergy as sources, and to elements that

dissipate energy as loads.

Standard symbols for a generalized source-and-load circuit are shown in Figure 2.7

which is symbolic representation of the circuit

shown in Figure 2.5.


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Ex 2.3: KVL For The Electric

Vehicle Battery Pack

Figure 2.8a depicts the battery pack in theSmokin Buckeye electric race car.

Apply KVL to the series connection of31,

12-V batteries that make up the batterysupply for the electric vehicle.

Figure 2.8(b) depicts the equivalent electricalcircuit, illustrating how the voltages supplied

by the battery are applied across the electricdrive that powers the vehicles 150-kW three-phase induction motor.


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The application ofKVL to the equivalent

circuit ofFigure 2.8(b) requires that:

VV

VV

VV

drive

nbattdrive

drive

n

batt

n

n

3721231

0

31

1

31

1


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2.3 Ideal Voltage And Current

Sources

Ideal Source: is a source that canprovide an arbitrary (constant) amount ofenergy.

There are two types of ideal sources:1. Voltage Sources.

2. Current Sources.

Dry-cell, alkaline, and lead-acid batteriesare all voltage sources (they are notideal, of course).


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You might have to think harder to come up

with a physical example that approximates

the behavior of an ideal current source.

However, For an ideal current source,assume a voltage source connected in series

with a circuit element that has a large

resistance to the flow of current from the

source provides a nearly constant though

smallcurrent and therefore acts very nearly

like an ideal current source.


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2.3.1 Ideal Voltage Sources

An ideal voltage source provides aprescribed voltage across its terminalsirrespective of the current flowing through

it. The amount of current supplied by the

source is determined by the circuitconnected to it.

Figure 2.9 depicts various symbols forvoltage sources.


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Note that the output voltage of an ideal source can be afunction of time.

Uppercase for DC, and lowercase for AC (time varying).

Figure 2.9 Ideal voltage sources


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By convention the direction of positivecurrent flow out of a voltage source is outof the positive terminal.

Figure 2.10 shows the source-loadrepresentations of an electrical circuit.

It depicts the connection of an energy sourcewith a passive circuit (i.e., a circuit that can

absorb and dissipate energy for example,the headlights and light bulb of our earlierexamples).


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2.3.2 Ideal Current Sources

An ideal current source

provides a prescribed

(constant) current to any

circuit connected to it.

To do so, it must be able

to generate

an arbitrary voltage

across its terminals..Figure 2.11 Symbol for

Ideal current source


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2.3.3 Dependent (Controlled)

Sources

The sources described so far have thecapability of generating a prescribed voltage

or current independent of any other element

within the circuit. Thus, they are termed Independent

Sources.

There exists another category of sources,however, whose output (current orvoltage)

is a function of some other voltage or

current in a circuit.


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These are called Dependent (orControlled) sources.

There are Two types of dependent voltage

source:1. Voltage controlled voltage source.

2. Current controlled voltage source.

There are two types of dependent currentsource:

1. Voltage controlled current source.

2. Current controlled current source.


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The symbols typically used to representdependent sources are depicted in Figure2.12

In Figure 2.12; the table illustrates therelationship between the source voltage orcurrent and the voltage or current itdepends on vxorix, respectively,which can be any voltage or current inthe circuit.


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2.4 Electric Power And Sign

Convention

Voltage: is the work per unit charge.

Power: is the work done per unit time.

Thus, the power,P

, eithergenerated ordissipated by a circuit element can be

represented by the following relationship:


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Thus, the electrical power generated by

an active element, or that dissipated or

stored by a passive element, is equal to

the product of the voltage across theelement and the current flowing

through it.

(joules/second, or watts)


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Power is a signed quantity (positive or

negative).

This distinction can be understood with

reference to Figure 2.13.

First, we need to assign the polarities of

the circuit elements, and the direction of

the current through these elements. The conventional way is shown in Figure

2.13, the procedure is as follow:


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Figure 2.13 The passive sign convention


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1. Choose an arbitrary direction of current flow.

2. Label polarities of all active elements

(voltage and current sources).

3. Current is usually leaving the positive

terminal of the source.

4. Assign polarities to all passive elements

(resistors and other loads); for passiveelements, current always flows into the

positive terminal.


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Thus, from figure 2.13, the polarity of the

voltage across the source and the direction of

the current through it indicate that the voltage

source is doing work in moving charge from alower potential to a higher potential (power is

generated).

On the other hand, the load is dissipatingpower, because the direction of the current

indicates that charge is being displaced from

a higher potential to a lower potential.


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The Passive Sign Convention: is thatthe power dissipated by a load is apositive quantity (or, conversely, that

the power generated by a source is apositive quantity).

In other words, if current flows from a

higher to a lower voltage (+ to ), thepower is dissipated and will be apositive quantity.

EX 2 4 U f th P i Si


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EX 2.4: Use of the Passive Sign

Convention

Apply the passive sign convention to the

circuit ofFigure 2.14.

Assume that the voltage drop across Load 1

is 8 V, that across Load 2 is 4 V; the currentin the circuit is 0.1 A.

Solution:

We can assign two differentdirections for the current as

shown in Figure 2.15.


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Current direction a) clockwise, b) counterclockwise


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This is the first step in the passive sign

convention.

According to each case, as a second step we

need to assign the polarities to each circuitelement.

Case 1: Assume clockwise current direction:

Thus, direction of the current is consistent with

the true polarity of the voltage source.

The source voltage will be a positive quantity.


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Since current flows from to + through thebattery, the power dissipated by thiselement will be a negative quantity:

PB= vB i= (12 V) (0.1 A) = 1.2 W

That is, the battery generates 1.2 W.

The power dissipated by the two loads willbe a positive quantity in both cases, since

current flows from + to :P1= v1 i= (8 V) (0.1 A) = 0.8 W

P2= v2 i= (4 V) (0.1 A) = 0.4 W


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Case 2: Assume counterclockwise currentdirection:

In this case, the current is not consistent with

the true polarity of the voltage source, thesource voltage will be a negative quantity.

Since current flows from + to through thebattery, the power dissipated by this element

will be a positive quantity; however, thesource voltage is a negative quantity:

PB= vB i= (12 V) (0.1 A) = 1.2 W


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That is, the battery generates 1.2 W, as in theprevious case.

The power dissipated by the two loads will bea positive quantity in both cases, since currentflows from + to :

P1= v1 i= (8 V) (0.1 A) = 0.8 W

P2= v2 i= (4 V) (0.1 A) = 0.4 W

Note that energy is conserved, as the sum of thepower dissipated by source and loads is zero.

Power supplied always equals power dissipated.

2 5 Ci it El t A d Th i


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2.5 Circuit Elements And Their

i-vCharacteristics

Figure 2.17 depicts a generalizedcircuit element: the variable iand vare the current throughand vol tageacrossthe element respectively.

If the voltage applied to the elementwere varied and the resulting currentmeasured, then, it is possible toconstruct a functional relationship

between voltage and current. This relation is called i-v

characteristic (volt-amperecharacteristic).


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Figure 2.18 depicts an experiment forempirically determining the i-vcharacteristic ofa tungsten filament light bulb.

A variable voltage source is used to apply

various voltages, and the current flowing throughthe element is measured for each appliedvoltage.

Hence, we can express the i-vcharacteristic of

a circuit element in functional form:i= f(v) or v= g(i)

Note that, since the bulb is passive element,hence, the power dissipated is positive.


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ContinueNote that, when a positive current

flows through the bulb, the voltage ispositive, and that, conversely, a

negative current flow corresponds to a

negative voltage.

In both cases the power dissipated by

the device is a positive quantity.


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The simplest form of the i-vcharacteristic

for a circuit element is a straight line:

i= kv, k is a constant

Figure 2.19 depicts the i-vcharacteristic ofan ideal voltage and current source.

An ideal voltage source generates a

prescribed voltage independent of the currentdrawn from the load.

The converse represent the current source.


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Figure 2.19i-vcharacteristics of ideal sources

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